0
votes
2answers
29 views

Symmetric Matrix Quadratic Form

Let $A,B\in\mathbb{M}_{n\times n}(\mathbb{R})$ and $A,B$ are symmetric matrics. Prove that if $\vec{x}^TA\vec{x} = \vec{x}^TB\vec{x}$ $\forall\vec{x}$, then $A=B$. Since $A,B$ are symmetric, they are ...
2
votes
1answer
52 views

When does $x^TAx + c^Tx$ have a global minimum?

This question is closely related to my last question about extended quadratic forms. I figured out a nice criterion, when $$f : \mathbb R^n \rightarrow \mathbb R$$ $$f(x) = x^TAx + c^Tx$$ has a ...
2
votes
1answer
74 views

Quadratic Form - New Axes = Eigenvectors of P, Order of Eigenvectors Important? [Kolman P539 Example 6]

Hypothesise that $P$ is the symmetric matrix of some quadratic form $g(\mathbf{ x} ) = \mathbf{ x^TAx} $. Then $P$ is the orthogonal matrix consisting of orthogonal eigenvectors of $A$. Moreover, use ...
2
votes
3answers
42 views

A is a matrix of positive defined quadratic form. How can I show, $A^{-1}$ is the same?

Let a square matrix A is a matrix of positive defined quadratic form. How can I show, that $A^{-1}$ also a matrix of a positive defined quadratic form? Positive defined quadratic form is A(x,y), that ...
0
votes
2answers
26 views

How to prove: a quadratic form with a matrix $ B = CC^T $ is positive defined?

Let a matrix $ C \in \Bbb K^{n \mathtt x n} : det(c) \ne 0 $ (K is any field - C or R) $ \Rightarrow $ a quadratic form with a matrix $ B = CC^T $ is positive defined one. How to prove it?
0
votes
1answer
37 views

Concavity of quadratic form

I know that the quadratic form $x'Ax$ is a concave in vector $x$ if matrix $A$ is negative semi definite. What happens if $A$ depends on $x$ (so that I have $x'A(x)x$), but I still know that $A(x)$ is ...
0
votes
0answers
13 views

Relaxed matrix factorization

I have an optimization problem like $ min~~ \frac{1}{2}||L||^2_F + \frac{1}{2}||R||^2_F,~~~~subject~to~ M=LR^T$ with respect to $L$ and $R$. I know that there are several factorizations that give ...
0
votes
1answer
18 views

Simplify quadratic polynomial with matrix

I am reading a paper and have trouble following equation (3): $$ (\mathbf{x}-\mathbf{d})^T \mathbf{A}_1 (\mathbf{x}-\mathbf{d}) + \mathbf{b}^T_1 (\mathbf{x}-\mathbf{d}) + c_1 = \\ \mathbf{x}^T ...
0
votes
1answer
23 views

Condition on the positivity of a quadratic form

We place ourself in $\mathbb{R}^{n}$. Let's consider a positive definite matrix $M \in \mathcal{M}_{n} (\mathbb{R})$, $V$ and $E$ $\in \mathbb{R}^{n}$, and $\alpha > 0$. We consider the ...
2
votes
2answers
68 views

Minimum of a quadratic form

If $\bf{A}$ is a real symmetric matrix, we know that it has orthogonal eigenvectors. Now, say we want to find a unit vector $\bf{n}$ that minimizes the form: $${\bf{n}}^T{\bf{A}}\ {\bf{n}}$$ How can ...
1
vote
1answer
27 views

completing the square for matrices

I'd like to calculate the posterior distribution given the prior distribution $w\sim N(0,\Sigma_p)$ and the likelihood $y|X,w\sim N(X^\top w,\sigma_n^2I).$ Ignoring everything that does not contain ...
-1
votes
1answer
58 views

Signature of quadratic form and eigenvalues

I'm asking about the signature of the quadratic form - the triple (n0, n+, n−). Is it true that n+ is the number of positive eigenvalues, and n- is the number of negative of eigenvalues of the matrix ...
0
votes
2answers
108 views

Find the symmetric matrix that represents the quadratic form $Q(X)=trace(X^2)$, $X\in mat_n\mathbb (R)$

as the title says, find the symmetric matrix (or signature) of $Q(X)=trace(X^2)$ where $X$ is an $n$ by $n$ matrix with real entries. the diagonal of $X^2$ is $$\sum_{k=1}^n x_{ik}x_{ki}$$ So ...
0
votes
1answer
57 views

Find the signature of the quadratic form

Very simple question but something doesn't make sense to me. We are given a quadratic form (bilinear map but on the same vector twice): $Q(v) = v^t *\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 ...
0
votes
1answer
65 views

Elliptical polarisation

In physic context one find the curve with parametrisation in t, $x=x_0\cos(t)$ and $y=y_0\cos(t+\varphi)$ with is an ellipse with equation ...
7
votes
2answers
265 views

How find this matrix $A=(\sqrt{i^2+j^2})$ eigenvalue

let the matrix $$A=(a_{ij})_{n\times n}$$ where $$a_{ij}=\sqrt{i^2+j^2}$$ Question: Find the difference $sign{(A)}$ can see this define:http://en.wikipedia.org/wiki/Sylvester's_law_of_inertia My ...
0
votes
0answers
39 views

Complete the square

How would I complete the square for $y$ after completing the square for x below. Note that $y,x$ are vectors and not scalars. $$ (y-A^Tx)^TC(y-A^Tx)+(x-\mu)^TD(x-\mu)\\ ...
0
votes
2answers
37 views

From the quadtratic form of a matrix to its symmetric matrix

I am trying to solve this quadratic form: $$ f(x_1 , x_2 , x_3) = x_1^2 + x_2^2 + 5x_3^2 -2x_1x_2 + 6x_1x_2+ 3x_2x_3$$ I know that the quadratic form is defined as: $$f(x)= x^t Qx$$ However my ...
0
votes
0answers
28 views

Squaring is not monotone [duplicate]

I know that the squaring of operators is not a monotone operation but I can't find a example showing exactly that. I'm looking for (preferably rather simple 2 by 2 matrices) $0<A\leq B$ such that ...
0
votes
1answer
65 views

bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose \begin{equation} \left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n. \end{equation} Can I ...
1
vote
3answers
130 views

A problem about the quadratic form $x^TAx=0$.

$A$ is an $m\times m$ real matrix in $\mathbb{R}^{m\times m}$. If $x^TAx=0\ \forall\ x\in\mathbb{R}^m$, can we conclude that $A=0_{m\times m}$? Why? Note: $x^T$ is the transpose of $x$. $0_{m\times ...
0
votes
1answer
63 views

Extremum of a multidimensional quadratic function

I have the following function: $$ g(h) = h'\Sigma\Sigma'h-h'm-r, $$ where $h$ is a vector in $\mathbb{R}^M$, $\Sigma$ is a $M\times K$ matrix such that $\Sigma\Sigma'$ is positive definite and has ...
4
votes
0answers
152 views

Proving the max of a quadratic form ${\mathbf x}^T\mathbf A \mathbf x$ can be attained when $x$ is from $n$-dimensional hypercube

updated: Maybe my original question is somewhat misleading. I rewrite some of the post. This is some research problem I'm working on. I have an $n\times n$ symmetric positive-definite matrix ...
0
votes
0answers
49 views

low rank decomposition of composite PSD matrix

The matrix $M=AVA^T - BCB^T +D$ is known to be positive semidefinite (PSD), where $V, C, D$ are each diagonal matrices with positive values, and $V, C$ has small size when compared to the size of $M$. ...
0
votes
1answer
104 views

Conditions for $v \otimes v$ to be positive semidefinite for complex $v$

I have a complex-symmetric matrix (in the sense $A=A^{T}$ not $A=A^{H}$), which is required to be positive semi-definite in the following sense (sometimes referred to as positive real): $ \Re(x^{*} A ...
2
votes
2answers
582 views

diagonalize quadratic form

I have this quadratic form $Q= x^2 + 4y^2 + 9z^2 + 4xy + 6xz+ 12yz$ And they ask me: for which values of $x,y$ and $z$ is $Q=0$? and I have to diagonalize also the quadratic form. I calculated ...
1
vote
1answer
45 views

How to show that $A=B-C$

How to show that for a real symmetric matrix $A,~A$ can be written as $A=B-C$ where $B,C$ are positive definite real symmetric matrices? Please help me ! I'm clueless.
1
vote
1answer
71 views

Solving quadratic form $\mathbf{x}^\mathrm{T}\mathbf{A}\mathbf{x} = c$ for $\mathbf{x}$

This is a simple question I hope, is there an easy way to solve: $$ \mathbf{x}^\mathrm{T}\mathbf{A}\mathbf{x} = c $$ for $\mathbf{x}$? (Assume $\mathbf{A}$ is positive definite). Geometrically the ...
6
votes
1answer
206 views

How to prove that $ E:=ABC D $ is also positive definite?

Now I think this is true: Let $A$, $B$, $C$ and $D$ be symmetric, positive definite matrices and suppose that $E:=ABCD $ is symmetric. How might I prove that $E$ is also positive definite? ...
1
vote
1answer
637 views

Real and complex canonical forms of quadratic form

How do I find the canonical form of $$q_1(x,y,z)= 4x^2 +4xz+2yz$$ Now I have put it in matrix form as: $$\left( \begin{matrix} 4 & 0 & 2 \\ 0 & 0 & 1 \\ ...
0
votes
1answer
71 views

Rewriting a quadratic Matrix equation as a quadratic vector equation

Consider the set of $N \times N$ matrices $\{W_i\}_{i=1}^{i=L}$, set of $N \times 1$vectors $\{g_i\}_{i=1}^{i=L}$ and $\{h_i\}_{i=1}^{i=L}$. Now consider the following sum \begin{align} ...
11
votes
3answers
222 views

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
1
vote
1answer
686 views

Why do we assume that a matrix in quadratic form is Symmetric?

I am looking to the review document for linear algebra and the part of the quadratic form (pg17) mentions about an assumption of being symmetric for a matrix in quadratic form. It also includes some ...
3
votes
0answers
106 views

counterexample for: integral forms that are equivalent as rational forms are also equivalent as integral forms

I have the feeling that I'm missing something very obvious: I'm looking for a counterexmple for the following statement for some $n>1$ (it is trivially true for $n=1$): Let $A,B\in\mathbb ...
0
votes
1answer
90 views

Need for algorithm on solving a set of quadratic matrix?

Firstly, I want to thank @adam W gives a good clue to solve my homework problem. I have a set of quadratic matrix need to solve(not one equation) according to the following form: ...
15
votes
5answers
678 views

Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$

Suppose $\vec x$ is a (non-zero) vector with integer coordinates in $\mathbb R^n$ such that $\|\vec x\| \in \mathbb Z$. Is it true that there is an orthogonal basis of $\mathbb R^n$ containing $\vec ...
1
vote
1answer
208 views

How do you construct a lattice from its basis or its Gram Matrix?

I'm really having trouble trying to understand this. A few weeks back, I got pretty interested in sphere packing and I'm trying to grasp the idea of using a matrix to represent the basis of a lattice. ...
1
vote
2answers
602 views

non-symmetric positive definite matrix!?

Is symmetry a necessary condition for positive (or negative) definiteness? If not: It can be proved that if $\mathbf{A}:(m\times m)$ is a square (non-symmetric) matrix, then $$ ...
0
votes
0answers
36 views

two non-degenerate quadratic forms on $GF(2)^2r$

I know this: There are two non-degenerate quadratic forms on $GF(2)^2r$. The hyperbolic form may be taken to be $Q^+(x)=x_0 x_1 + \cdots +x_{2r-2}x_{2r-1}$ , and the elliptic form to be ...
4
votes
1answer
226 views

Matrix Equation with Quadratic form

I am working in a problem that involves multivariate normal distributions and, at a given point, I need to solve the following matrix equation: $$x=\sqrt{x^{\prime}\Sigma^{-1}x} \cdot y$$ Where $x$ ...
2
votes
2answers
3k views

Derivative of Quadratic Form

For the Quadratic Form $X^TAX; X\in\mathbb{R}^n, A\in\mathbb{R}^{n \times n}$ (which simplifies to $\Sigma_{i=0}^n\Sigma_{j=0}^nA_{ij}x_ix_j$), I tried to take the derivative wrt. X ($\Delta_X X^TAX$) ...
1
vote
7answers
352 views

positive definite quadratic form

Is $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ positive definite? Approach: The matrix of this quadratic form can be derived to be the following $$M := \begin{pmatrix} 1 & ...
0
votes
1answer
72 views

Action of $SL_2(Z)$ on Markoff quadratic forms

My setting is as follows: Fix a Markoff form $f_m(x,y)$ (see definition in the link below). If $f_m$ has the form ${\alpha}x^2+{\beta}xy+{\gamma}y^2$ then each element $A\in SL_2(Z)$ acts on $f_m$ in ...
1
vote
2answers
482 views

Computing the rank and signature of a quadratic form - quick way?

Is there a 'quick way' of computing the rank and signature of the quadratic form $$q(x,y,z) = xy - xz$$ as I can only think of doing the huge computation where you find a basis such that the matrix of ...
2
votes
1answer
5k views

Quadratic equation -> matrix?

Problem: Find the EigenValues and EigenVectors of the matrix associated with quadratic forms $2x^2+6y^2+2z^2+8xz$. I know how to convert a set of polynomial equations to a matrix but I have no clue ...
0
votes
1answer
450 views

Canonical form of a Matrix

My understanding of canonical form is very limited, and so may require some help. Suppose a quadratic of the form: $$ x_1*x_2+x_1*x_3=Q.$$ How would one go about putting that into canonical form, ...
0
votes
1answer
973 views

Proving that a matrix is negative definite using its principal minors

I am interested to find out the proof for the following statement (it's from my textbook and it is stated without proof): A symmetric matrix is negative definite if and only if all of its ...
4
votes
1answer
444 views

When are two diagonal matrices congruent?

This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...
2
votes
3answers
515 views

positive symmetric matrices and positive-definiteness

Is a symmetric real matrix with diagonal entries strictly greater than 1 and off-diagonal entries positive but strictly less than 1 necessarily positive-semidefinite?